In their seminal paper [1]} Hochman and Shmerkin proved the Furstenberg's sumset conjecture, that for two closed subsets \(X,Y\)  of the unit circle \(\mathbb {T}=[0,1] \mod {1}\)  that are \(\times 2\)  and \(\times 3\)  invariant respectively, their summation has the largest possible dimension. More precisely, they showed that for all \(s\ne 0\) ,
\(\dim (X+sY)=\min \lbrace 1,\dim X+\dim Y\rbrace ,\) 
